label ith. They should also label the two basesb 1 andb 2.
b. Now they can trace and cut out a second congruent trapezoid and label it as they did the first.
c. The two trapezoids can be arranged into a parallelogram and glued down to another piece of paper.
d. Identify the base and height of the parallelogram in terms of the trapezoid variables. Then substitute these
expressions into the area formula of a parallelogram to derive the area formula for a trapezoid.
e. Remember that two congruent trapezoids were used in the parallelogram, and the formula should only find the
area of one trapezoid.Kite
a. Have the students draw a kite. They should start by making perpendicular diagonals, one of which is bisecting
the other. Then they can connect the vertices to form a kite.
b. Now they can draw in the rectangle around the kite.
c. Identify the base and height of the parallelogram in terms ofd 1 andd 2 , and then substitute into the parallelo-
gram area formula to derive the kite area formula.
d. Now have the students cut off the four triangles that are not part of the kite and arrange them over the congruent
triangle in the kite to demonstrate that the area of the kite is half the area of the rectangle.Rhombus
The area of a rhombus can be found using either the kite or parallelogram area formulas. Use this as an opportunity
to review subsets and what they mean in terms of applying formulas and theorems.
Areas of Similar Polygons
Adjust the Scale Factor -It is difficult for students to remember to square and cube the scale factor when writing
proportions involving area and volume. Writing and solving a proportion is a skill they know well and have used
frequently. Once the process is started, it is hard to remember to add that extra step of checking and adjusting the
scale factor in the middle of the process. Here are some ways to reinforce this step in the students’ minds.
a. Inform students that this material is frequently used on the SAT and other standardized tests in some of the
more difficult problems.
b. Play with graph paper. Have students draw similar shape on graph paper. They can estimate the area by
counting squares, and then compare the ratio of the areas to the ratio of the side lengths. Creating the shapes
on graph paper will give the students a good visual impression of the areas.
c. Write out steps, or have the students write out the process they will use to tackle these problems. (1) Write a
ratio comparing the two polygons. (2) Identify the type of ratio: linear, area, or volume. (3) Adjust the ratio
using powers or roots to get the desired ratio. (4) Write and solve a proportion.
d. Mix-up the exercises so that students will have to square the ratio in one problem and not in the next. Keep
them on the lookout. Make them analyze the situation instead of falling into a habit.Additional Exercises:
- The ratio of the lengths of the sides of two squares is 1 : 2. What is the ratio of their areas?
Answer: 1 : 4
- The area of a small triangle is 15 cm^2 , and has a height of 5 cm. A larger similar triangle has an area of 60 cm^2.
What is the corresponding height of the larger triangle?
Answer:
area ratio 15 : 60 or 1 : 4 height linear ratio 1 : 2=10 cm
Chapter 2. Geometry TE - Common Errors